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Theorem sbalex 2236
Description: Equivalence of two ways to express proper substitution of a setvar for another setvar disjoint from it in a formula. This proof of their equivalence does not use df-sb 2069.

That both sides of the biconditional express proper substitution is proved by sb5 2268 and sb6 2089. The implication "to the left" is equs4v 2004 and does not require ax-10 2138 nor ax-12 2172. It also holds without disjoint variable condition if we allow more axioms (see equs4 2415). Theorem 6.2 of [Quine] p. 40. Theorem equs5 2459 replaces the disjoint variable condition with a distinctor antecedent. Theorem equs45f 2458 replaces the disjoint variable condition on 𝑥, 𝑡 with the nonfreeness hypothesis of 𝑡 in 𝜑. (Contributed by NM, 14-Apr-2008.) Revised to use equsexv 2260 in place of equsex 2417 in order to remove dependency on ax-13 2371. (Revised by BJ, 20-Dec-2020.) Revise to remove dependency on df-sb 2069. (Revised by BJ, 21-Sep-2024.)

Assertion
Ref Expression
sbalex (∃𝑥(𝑥 = 𝑡𝜑) ↔ ∀𝑥(𝑥 = 𝑡𝜑))
Distinct variable group:   𝑥,𝑡
Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem sbalex
StepHypRef Expression
1 nfa1 2149 . . 3 𝑥𝑥(𝑥 = 𝑡𝜑)
2 ax12v2 2174 . . . 4 (𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))
32imp 408 . . 3 ((𝑥 = 𝑡𝜑) → ∀𝑥(𝑥 = 𝑡𝜑))
41, 3exlimi 2211 . 2 (∃𝑥(𝑥 = 𝑡𝜑) → ∀𝑥(𝑥 = 𝑡𝜑))
5 equs4v 2004 . 2 (∀𝑥(𝑥 = 𝑡𝜑) → ∃𝑥(𝑥 = 𝑡𝜑))
64, 5impbii 208 1 (∃𝑥(𝑥 = 𝑡𝜑) ↔ ∀𝑥(𝑥 = 𝑡𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540  wex 1782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138  ax-12 2172
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787
This theorem is referenced by:  equsexvOLD  2261  sb5  2268  dfsb7  2276  mopick  2626  alexeqg  3606  dfdif3  4079  pm13.196a  42768
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